Abstract
Historical numerical methods for solving stiff ordinary differential equations are investigated and a new numerical framework developed and applied to a variety of hydrodynamic stability problems where the effects of passive wall compliance are investigated. The compound matrix method is set in a general coordinate free framework using exterior algebra, and is considered to be the most accurate and easy to implement method for complex systems. The effect of passive wall compliance on the Blasius boundary layer flow is studied. The linear stability of the mean flow state is considered using the new numerical framework and shooting technique. A Newton algorithm is implemented to converge the eigenvalue such that the boundary condition at the surface is satisfied. Curves of neutral stability are produced in the Re-alpha plane for various degrees of compliance, damping and tension parameters. Three dimensionality is incorporated first by a simple introduction of rotation in the flow leading to the investigation of the Ekman boundary layer problem where a Coriolis force instability mechanism (type-2) produces streamwise rolls at modest Reynolds numbers. The linear stability of the Ekman mean flow state is considered using an extension of the new numerical framework for use on the vector space Lambda(3)(C6). Curves of neutral stability are produced in the E-gamma plane for constant values of the Reynolds number and in the Re-gamma plane for a selection of constant angle of orientation, E. This work is extended to consider the effects of wall compliance on the type-2 viscous instability mechanism, with the type-1 mode of instability briefly discussed. Three-dimensionality is then used for a direct application to the dolphin. The stability of the attachment-line boundary layer is investigated on flows past swept wings, relating directly to the dolphins swept-back fins. Wall compliance, modelling the dolphins skin, is included for analysis of its effect on the attachment-line instability.